Lecture 3. Linear Programming. 3B1B Optimization Michaelmas 2015 A. Zisserman. Extreme solutions. Simplex method. Interior point method

Size: px
Start display at page:

Download "Lecture 3. Linear Programming. 3B1B Optimization Michaelmas 2015 A. Zisserman. Extreme solutions. Simplex method. Interior point method"

Transcription

1 Lecture 3 3B1B Optimization Michaelmas 2015 A. Zisserman Linear Programming Extreme solutions Simplex method Interior point method Integer programming and relaxation

2 The Optimization Tree

3 Linear Programming The name is historical, it should really be called Linear Optimization. The problem consists of three parts: A linear function to be maximized maximize f(x) = c 1 x 1 + c 2 x c n x n Problem constraints Non-negative variables subject to a 11 x 1 + a 12 x a 1n x n < b 1 a 21 x 1 + a 22 x a 2n x n < b 2 a m1 x 1 + a m2 x a mn x n < b m e.g. x 1, x 2 > 0

4 Linear Programming The problem is usually expressed in matrix form and then it becomes: Maximize c > x subject to Ax b, x 0 where A is a m n matrix. The objective function, c > x,andconstraints,ax b, are all linear functions of x. (Linear programming problems are convex, so a local optimum is the global optimum.)

5 Example 1 Suppose that a farmer has a piece of farm land, say A square kilometers large, to be planted with either wheat or barley or some combination of the two. The farmer has a limited permissible amount F of fertilizer and P of insecticide which can be used, each of which is required in different amounts per unit area for wheat (F 1, P 1 ) and barley (F 2, P 2 ). Let S 1 be the selling price of wheat, and S 2 the price of barley, and denote the area planted with wheat and barley as x 1 and x 2 respectively. The optimal number of square kilometers to plant with wheat vs. barley can be expressed as a linear programming problem. x 1 x 2 A

6 Example 1 cont. Maximize S 1 x 1 + S 2 x 2 (the revenue this is the objective function ) subject to x 1 +x 2 < A (limit on total area) F 1 x 1 + F 2 x 2 < F (limit on fertilizer) P 1 x 2 + P 2 x 2 < P (limit on insecticide) x 1 >= 0, x 2 > 0 (cannot plant a negative area) which in matrix form becomes maximize subject to

7 Example 2: Max flow Given: a weighted directed graph, source s, destination t Interpret edge weights as capacities Goal: Find maximum flow from s to t Flow does not exceed capacity in any edge Flow at every vertex satisfies equilibrium [ flow in equals flow out ] e.g. oil flowing through pipes, internet routing

8 Example 3 cont. Slide: Robert Sedgewick and Kevin Wayne

9 Example 2 cont.

10 Example 3: shortest path Given: a weighted directed graph, with a single source s Distance from s to v: length of the shortest part from s to v Goal: Find distance (and shortest path) to every vertex e.g. plotting routes on Google maps

11 Application Minimize number of stops (lengths = 1) Minimize amount of time (positive lengths)

12 LP why is it important? We have seen examples of: allocating limited resources network flow shortest path Others include: matching assignment It is a widely applicable problem-solving model because: non-negativity is the usual constraint on any variable that represents an amount of something one is often interested in bounds imposed by limited resources. Dominates world of industry

13 Linear Programming 2D example Cost function: f(x 1,x 2 ) = 0x x 2 Inequality constraints: x 1 + x 2 2 x x x 2 4 max f(x x 1,x 1,x 2 ) 2 x 1,x 2 0

14 Feasible Region Inequality constraints: x x 1 + x 2 =2 x 1 =1.5 x 1 + x 2 2 x x x 2 4 x 1,x x x 2 = x 1

15 LP example max f(x x 1,x 1,x 2 ) 2 Cost function: z z =0.5x 2 f(x 1,x 2 ) = 0x x 2 Inequality constraints: x 1 + x 2 2 x feasible region x 2 x x 2 4 x 1,x 2 0 x 1 extreme point

16 LP example max f(x x 1,x 1,x 2 ) 2 x extreme point contour lines Cost function: 1.6 f(x 1,x 2 ) = 0x x 2 Inequality constraints: x 1 + x 2 2 x x x x 1,x feasible region x 1

17 LP example change of cost function max f(x x 1,x 1,x 2 ) 2 Cost function: f(x 1,x 2 ) = c > x = c 1 x 1 + c 2 x 2 x extreme point contour lines Inequality constraints: x 1 + x 2 2 x x x c x 1,x feasible region x 1

18 Linear Programming optima at vertices The key point is that for any (linear) objective function the optima only occur at the corners (vertices) of the feasible polygonal region (never on the interior region). Similarly, in 3D the optima only occur at the vertices of a polyhedron (and in nd at the vertices of a polytope). However, the optimum is not necessarily unique: it is possible to have a set of optimal solutions covering an edge or face of a polyhedron.

19 Cf Constrained Quadratic Optimization z extreme point 5 x 2 0 x feasible region

20 Linear Programming solution idea So, even though LP is formulated in terms of continuous variables, x, there are only a finite number of discrete possible solutions that need be explored. How to find the maximum?? Try every vertex? But there are too many in large problems. Instead, simply go from one vertex to the next increasing the cost function each time, and in an intelligent manner to avoid having to visit (and test) every vertex. This is the idea of the simplex algorithm

21 Sketch solutions for optimization methods We will look at 2 methods of solution: 1. Simplex method Tableau exploration of vertices based on linear algebra 2. Interior point method Continuous optimization with constraints cast as barriers

22 1. Simplex method Optimum must be at the intersection of constraints Intersections are easy to find, change inequalities to equalities Intersections are called basic solutions 2 x 1 =1.5 some intersections are outside the feasible region (e.g. f) and so need not be considered the others (which are vertices of the feasible region) are called basic feasible solutions a x 1 + x 2 =2 b f x x 2 =4 0.6 c e d

23 Worst complexity There are Cn m m! = (m n)! n! possible solutions, where m is the total number of constraints and n the dimension of the space a x 1 + x 2 =2 x 1 = In this case C 5 2 = 5! (3)! 2! = b f x x 2 =4 However, for large problems the number of solutions can be huge, and it is not realistic to explore them all e c d

24 The Simplex Algorithm Start from a basic feasible solution (i.e. a vertex of feasible region) Consider all the vertices connected to the current one by an edge Choose the vertex which increases the cost function the most (and is still feasible of course) Repeat until no further increases are possible In practice this is very efficient and avoids visiting all vertices

25 Matlab LP Function linprog Linprog() for medium scale problems uses the Simplex algorithm Example Find x that minimizes f(x) = 5x 1 4x 2 6x 3, subject to x 1 x 2 + x x 1 + 2x 2 + 4x x 1 + 2x x 1, 0 x 2, 0 x 3. >> f = [-5; -4; -6]; >> A = [ ]; >> b = [20; 42; 30]; >> lb = zeros(3,1); >> x = linprog(f,a,b,[],[],lb); >> Optimization terminated. >> x x =

26 MOSEK For large scale linear and quadratic optimization problems

27 2. Interior Point Method Solve LP using continuous optimization methods Represent inequalities by barrier functions Follow path through interior of feasible region to vertex c

28 Barrier function method We wish to solve the following problem Minimize subject to f(x) =c > x a i > x b i,i=1...m Problem could be rewritten as Minimize f(x)+ mx i=1 I(a i > x b i ) where I is the indicator function I(u) = 0 for u 0 = for u>0 The difficulty here is that I(u) is not differentiable.

29 Approximation via logarithmic barrier functon Approximate indicator function by logarithmic barrier Minimize f(x) 1 t mx i=1 log( a i > x + b i ) for t>0, (1/t)log( u) is a smooth approximation of I(u) approximation improves as t log( u) t

30 Minimizing function for different Values of t. Function f(x) to be minimized subject to x 1, x 1 Minima converge to the constrained minimum. Function tf(x) log(1 x 2 ) for increasing values of t.

31 Algorithm for Interior Point Method Problem becomes Algorithm Minimize tf(x) mx i=1 log( a i > x + b i ) Solve using Newton s method t μt repeat until convergence As t increases this converges to the solution of the original problem

32 Example Diagram copied from Boyd and Vandenberghe

33 Integer programming There are often situations where the solution is required to be an integer or have boolean values (0 or 1 only). For example: assignment problems scheduling problems matching problems Linear programming can also be used for these cases

34 Example: assignment problem agents tasks Objective: assign n agents to n tasks to minimize total cost Assignment constraints: each agent assigned to one task only each task assigned to one agent only

35 Example: assignment problem agents tasks Objective: assign n agents to n tasks to minimize total cost Assignment constraints: each agent assigned to one task only each task assigned to one agent only

36 cost matrix tasks tasks a g e n t s a g e n t s

37 Example: assignment problem Problem specification x ij is the assignment of agent i to task j (can take values 0 or 1) c ij is the (non-negative) cost of assigning agent i to task j Minimize total cost f(x) = P ij x ij c ij over the n 2 variables x ij tasks j Example solution for x ij each agent i assigned to one task only only one entry in each row each task j assigned to one agent only only one entry in each column a g e n t s i

38 Example: assignment problem Linear Programme formulation min x ij subject to (inequalities) (equalities) f(x) = X ij X j X i x ij c ij tasks j x ij 1, i each agent assigned to at most one task x ij =1, j each task assigned to exactly one agent This is a relaxation of the problem because the variables x ij are not forced to take boolean values. However, it can be shown that the solution x ij only takes the values 0 or 1 a g e n t s i

39 Agents to tasks assignment problem Optimal agents to tasks assignment cost: agents tasks agents tasks Cost of assignment = distance between agent and task e.g. application: tracking, correspondence

40 Example application: tracking pedestrians Multiple Object Tracking using Flow Linear Programming, Berclaz, Fleuret & Fua, 2009

41 What is next? Convexity (when does a function have only a global minimum?) Robust cost functions Stochastic algorithms

Linear Programming. Widget Factory Example. Linear Programming: Standard Form. Widget Factory Example: Continued.

Linear Programming. Widget Factory Example. Linear Programming: Standard Form. Widget Factory Example: Continued. Linear Programming Widget Factory Example Learning Goals. Introduce Linear Programming Problems. Widget Example, Graphical Solution. Basic Theory:, Vertices, Existence of Solutions. Equivalent formulations.

More information

Linear Programming: Introduction

Linear Programming: Introduction Linear Programming: Introduction Frédéric Giroire F. Giroire LP - Introduction 1/28 Course Schedule Session 1: Introduction to optimization. Modelling and Solving simple problems. Modelling combinatorial

More information

The Simplex Method. yyye

The Simplex Method.  yyye Yinyu Ye, MS&E, Stanford MS&E310 Lecture Note #05 1 The Simplex Method Yinyu Ye Department of Management Science and Engineering Stanford University Stanford, CA 94305, U.S.A. http://www.stanford.edu/

More information

Chapter 15 Introduction to Linear Programming

Chapter 15 Introduction to Linear Programming Chapter 15 Introduction to Linear Programming An Introduction to Optimization Spring, 2014 Wei-Ta Chu 1 Brief History of Linear Programming The goal of linear programming is to determine the values of

More information

Lecture 4 Linear Programming Models: Standard Form. August 31, 2009

Lecture 4 Linear Programming Models: Standard Form. August 31, 2009 Linear Programming Models: Standard Form August 31, 2009 Outline: Lecture 4 Standard form LP Transforming the LP problem to standard form Basic solutions of standard LP problem Operations Research Methods

More information

Linear Programming Notes V Problem Transformations

Linear Programming Notes V Problem Transformations Linear Programming Notes V Problem Transformations 1 Introduction Any linear programming problem can be rewritten in either of two standard forms. In the first form, the objective is to maximize, the material

More information

2.3 Convex Constrained Optimization Problems

2.3 Convex Constrained Optimization Problems 42 CHAPTER 2. FUNDAMENTAL CONCEPTS IN CONVEX OPTIMIZATION Theorem 15 Let f : R n R and h : R R. Consider g(x) = h(f(x)) for all x R n. The function g is convex if either of the following two conditions

More information

Lecture 2: August 29. Linear Programming (part I)

Lecture 2: August 29. Linear Programming (part I) 10-725: Convex Optimization Fall 2013 Lecture 2: August 29 Lecturer: Barnabás Póczos Scribes: Samrachana Adhikari, Mattia Ciollaro, Fabrizio Lecci Note: LaTeX template courtesy of UC Berkeley EECS dept.

More information

Section Notes 3. The Simplex Algorithm. Applied Math 121. Week of February 14, 2011

Section Notes 3. The Simplex Algorithm. Applied Math 121. Week of February 14, 2011 Section Notes 3 The Simplex Algorithm Applied Math 121 Week of February 14, 2011 Goals for the week understand how to get from an LP to a simplex tableau. be familiar with reduced costs, optimal solutions,

More information

Using the Simplex Method in Mixed Integer Linear Programming

Using the Simplex Method in Mixed Integer Linear Programming Integer Using the Simplex Method in Mixed Integer UTFSM Nancy, 17 december 2015 Using the Simplex Method in Mixed Integer Outline Mathematical Programming Integer 1 Mathematical Programming Optimisation

More information

SOLVING LINEAR SYSTEM OF INEQUALITIES WITH APPLICATION TO LINEAR PROGRAMS

SOLVING LINEAR SYSTEM OF INEQUALITIES WITH APPLICATION TO LINEAR PROGRAMS SOLVING LINEAR SYSTEM OF INEQUALITIES WITH APPLICATION TO LINEAR PROGRAMS Hossein Arsham, University of Baltimore, (410) 837-5268, harsham@ubalt.edu Veena Adlakha, University of Baltimore, (410) 837-4969,

More information

Linear Programming I

Linear Programming I Linear Programming I November 30, 2003 1 Introduction In the VCR/guns/nuclear bombs/napkins/star wars/professors/butter/mice problem, the benevolent dictator, Bigus Piguinus, of south Antarctica penguins

More information

Outline. Linear Programming (LP): Simplex Search. Simplex: An Extreme-Point Search Algorithm. Basic Solutions

Outline. Linear Programming (LP): Simplex Search. Simplex: An Extreme-Point Search Algorithm. Basic Solutions Outline Linear Programming (LP): Simplex Search Benoît Chachuat McMaster University Department of Chemical Engineering ChE 4G03: Optimization in Chemical Engineering 1 Basic Solutions

More information

Linear Programming for Optimization. Mark A. Schulze, Ph.D. Perceptive Scientific Instruments, Inc.

Linear Programming for Optimization. Mark A. Schulze, Ph.D. Perceptive Scientific Instruments, Inc. 1. Introduction Linear Programming for Optimization Mark A. Schulze, Ph.D. Perceptive Scientific Instruments, Inc. 1.1 Definition Linear programming is the name of a branch of applied mathematics that

More information

By W.E. Diewert. July, Linear programming problems are important for a number of reasons:

By W.E. Diewert. July, Linear programming problems are important for a number of reasons: APPLIED ECONOMICS By W.E. Diewert. July, 3. Chapter : Linear Programming. Introduction The theory of linear programming provides a good introduction to the study of constrained maximization (and minimization)

More information

Chap 4 The Simplex Method

Chap 4 The Simplex Method The Essence of the Simplex Method Recall the Wyndor problem Max Z = 3x 1 + 5x 2 S.T. x 1 4 2x 2 12 3x 1 + 2x 2 18 x 1, x 2 0 Chap 4 The Simplex Method 8 corner point solutions. 5 out of them are CPF solutions.

More information

3y 1 + 5y 2. y 1 + y 2 20 y 1 0, y 2 0.

3y 1 + 5y 2. y 1 + y 2 20 y 1 0, y 2 0. 1 Linear Programming A linear programming problem is the problem of maximizing (or minimizing) a linear function subject to linear constraints. The constraints may be equalities or inequalities. 1.1 Example

More information

Chapter 6. Linear Programming: The Simplex Method. Introduction to the Big M Method. Section 4 Maximization and Minimization with Problem Constraints

Chapter 6. Linear Programming: The Simplex Method. Introduction to the Big M Method. Section 4 Maximization and Minimization with Problem Constraints Chapter 6 Linear Programming: The Simplex Method Introduction to the Big M Method In this section, we will present a generalized version of the simplex method that t will solve both maximization i and

More information

LINEAR PROGRAMMING PROBLEM: A GEOMETRIC APPROACH

LINEAR PROGRAMMING PROBLEM: A GEOMETRIC APPROACH 59 LINEAR PRGRAMMING PRBLEM: A GEMETRIC APPRACH 59.1 INTRDUCTIN Let us consider a simple problem in two variables x and y. Find x and y which satisfy the following equations x + y = 4 3x + 4y = 14 Solving

More information

Nonlinear Optimization: Algorithms 3: Interior-point methods

Nonlinear Optimization: Algorithms 3: Interior-point methods Nonlinear Optimization: Algorithms 3: Interior-point methods INSEAD, Spring 2006 Jean-Philippe Vert Ecole des Mines de Paris Jean-Philippe.Vert@mines.org Nonlinear optimization c 2006 Jean-Philippe Vert,

More information

3.1 Solving Systems Using Tables and Graphs

3.1 Solving Systems Using Tables and Graphs Algebra 2 Chapter 3 3.1 Solve Systems Using Tables & Graphs 3.1 Solving Systems Using Tables and Graphs A solution to a system of linear equations is an that makes all of the equations. To solve a system

More information

Quiz 1 Sample Questions IE406 Introduction to Mathematical Programming Dr. Ralphs

Quiz 1 Sample Questions IE406 Introduction to Mathematical Programming Dr. Ralphs Quiz 1 Sample Questions IE406 Introduction to Mathematical Programming Dr. Ralphs These questions are from previous years and should you give you some idea of what to expect on Quiz 1. 1. Consider the

More information

Good luck, veel succes!

Good luck, veel succes! Final exam Advanced Linear Programming, May 7, 13.00-16.00 Switch off your mobile phone, PDA and any other mobile device and put it far away. No books or other reading materials are allowed. This exam

More information

Can linear programs solve NP-hard problems?

Can linear programs solve NP-hard problems? Can linear programs solve NP-hard problems? p. 1/9 Can linear programs solve NP-hard problems? Ronald de Wolf Linear programs Can linear programs solve NP-hard problems? p. 2/9 Can linear programs solve

More information

Linear Programming, Lagrange Multipliers, and Duality Geoff Gordon

Linear Programming, Lagrange Multipliers, and Duality Geoff Gordon lp.nb 1 Linear Programming, Lagrange Multipliers, and Duality Geoff Gordon lp.nb 2 Overview This is a tutorial about some interesting math and geometry connected with constrained optimization. It is not

More information

3. Evaluate the objective function at each vertex. Put the vertices into a table: Vertex P=3x+2y (0, 0) 0 min (0, 5) 10 (15, 0) 45 (12, 2) 40 Max

3. Evaluate the objective function at each vertex. Put the vertices into a table: Vertex P=3x+2y (0, 0) 0 min (0, 5) 10 (15, 0) 45 (12, 2) 40 Max SOLUTION OF LINEAR PROGRAMMING PROBLEMS THEOREM 1 If a linear programming problem has a solution, then it must occur at a vertex, or corner point, of the feasible set, S, associated with the problem. Furthermore,

More information

Graphical method. plane. (for max) and down (for min) until it touches the set of feasible solutions. Graphical method

Graphical method. plane. (for max) and down (for min) until it touches the set of feasible solutions. Graphical method The graphical method of solving linear programming problems can be applied to models with two decision variables. This method consists of two steps (see also the first lecture): 1 Draw the set of feasible

More information

Optimization Modeling for Mining Engineers

Optimization Modeling for Mining Engineers Optimization Modeling for Mining Engineers Alexandra M. Newman Division of Economics and Business Slide 1 Colorado School of Mines Seminar Outline Linear Programming Integer Linear Programming Slide 2

More information

Lecture 3: Linear Programming Relaxations and Rounding

Lecture 3: Linear Programming Relaxations and Rounding Lecture 3: Linear Programming Relaxations and Rounding 1 Approximation Algorithms and Linear Relaxations For the time being, suppose we have a minimization problem. Many times, the problem at hand can

More information

Linear Programming. March 14, 2014

Linear Programming. March 14, 2014 Linear Programming March 1, 01 Parts of this introduction to linear programming were adapted from Chapter 9 of Introduction to Algorithms, Second Edition, by Cormen, Leiserson, Rivest and Stein [1]. 1

More information

I. Solving Linear Programs by the Simplex Method

I. Solving Linear Programs by the Simplex Method Optimization Methods Draft of August 26, 2005 I. Solving Linear Programs by the Simplex Method Robert Fourer Department of Industrial Engineering and Management Sciences Northwestern University Evanston,

More information

Max Flow. Lecture 4. Optimization on graphs. C25 Optimization Hilary 2013 A. Zisserman. Max-flow & min-cut. The augmented path algorithm

Max Flow. Lecture 4. Optimization on graphs. C25 Optimization Hilary 2013 A. Zisserman. Max-flow & min-cut. The augmented path algorithm Lecture 4 C5 Optimization Hilary 03 A. Zisserman Optimization on graphs Max-flow & min-cut The augmented path algorithm Optimization for binary image graphs Applications Max Flow Given: a weighted directed

More information

Definition of a Linear Program

Definition of a Linear Program Definition of a Linear Program Definition: A function f(x 1, x,..., x n ) of x 1, x,..., x n is a linear function if and only if for some set of constants c 1, c,..., c n, f(x 1, x,..., x n ) = c 1 x 1

More information

What is Linear Programming?

What is Linear Programming? Chapter 1 What is Linear Programming? An optimization problem usually has three essential ingredients: a variable vector x consisting of a set of unknowns to be determined, an objective function of x to

More information

5 INTEGER LINEAR PROGRAMMING (ILP) E. Amaldi Fondamenti di R.O. Politecnico di Milano 1

5 INTEGER LINEAR PROGRAMMING (ILP) E. Amaldi Fondamenti di R.O. Politecnico di Milano 1 5 INTEGER LINEAR PROGRAMMING (ILP) E. Amaldi Fondamenti di R.O. Politecnico di Milano 1 General Integer Linear Program: (ILP) min c T x Ax b x 0 integer Assumption: A, b integer The integrality condition

More information

Introduction to Linear Programming.

Introduction to Linear Programming. Chapter 1 Introduction to Linear Programming. This chapter introduces notations, terminologies and formulations of linear programming. Examples will be given to show how real-life problems can be modeled

More information

Geometry of Linear Programming

Geometry of Linear Programming Chapter 2 Geometry of Linear Programming The intent of this chapter is to provide a geometric interpretation of linear programming problems. To conceive fundamental concepts and validity of different algorithms

More information

Discrete Optimization

Discrete Optimization Discrete Optimization [Chen, Batson, Dang: Applied integer Programming] Chapter 3 and 4.1-4.3 by Johan Högdahl and Victoria Svedberg Seminar 2, 2015-03-31 Todays presentation Chapter 3 Transforms using

More information

Basic Components of an LP:

Basic Components of an LP: 1 Linear Programming Optimization is an important and fascinating area of management science and operations research. It helps to do less work, but gain more. Linear programming (LP) is a central topic

More information

1 Introduction. Linear Programming. Questions. A general optimization problem is of the form: choose x to. max f(x) subject to x S. where.

1 Introduction. Linear Programming. Questions. A general optimization problem is of the form: choose x to. max f(x) subject to x S. where. Introduction Linear Programming Neil Laws TT 00 A general optimization problem is of the form: choose x to maximise f(x) subject to x S where x = (x,..., x n ) T, f : R n R is the objective function, S

More information

Linear Programming Problems

Linear Programming Problems Linear Programming Problems Linear programming problems come up in many applications. In a linear programming problem, we have a function, called the objective function, which depends linearly on a number

More information

Chapter 5. Linear Inequalities and Linear Programming. Linear Programming in Two Dimensions: A Geometric Approach

Chapter 5. Linear Inequalities and Linear Programming. Linear Programming in Two Dimensions: A Geometric Approach Chapter 5 Linear Programming in Two Dimensions: A Geometric Approach Linear Inequalities and Linear Programming Section 3 Linear Programming gin Two Dimensions: A Geometric Approach In this section, we

More information

1 Solving LPs: The Simplex Algorithm of George Dantzig

1 Solving LPs: The Simplex Algorithm of George Dantzig Solving LPs: The Simplex Algorithm of George Dantzig. Simplex Pivoting: Dictionary Format We illustrate a general solution procedure, called the simplex algorithm, by implementing it on a very simple example.

More information

SUPPLEMENT TO CHAPTER

SUPPLEMENT TO CHAPTER SUPPLEMENT TO CHAPTER 6 Linear Programming SUPPLEMENT OUTLINE Introduction and Linear Programming Model, 2 Graphical Solution Method, 5 Computer Solutions, 14 Sensitivity Analysis, 17 Key Terms, 22 Solved

More information

Introduction to Linear Programming (LP) Mathematical Programming (MP) Concept

Introduction to Linear Programming (LP) Mathematical Programming (MP) Concept Introduction to Linear Programming (LP) Mathematical Programming Concept LP Concept Standard Form Assumptions Consequences of Assumptions Solution Approach Solution Methods Typical Formulations Massachusetts

More information

Lecture 1: Linear Programming Models. Readings: Chapter 1; Chapter 2, Sections 1&2

Lecture 1: Linear Programming Models. Readings: Chapter 1; Chapter 2, Sections 1&2 Lecture 1: Linear Programming Models Readings: Chapter 1; Chapter 2, Sections 1&2 1 Optimization Problems Managers, planners, scientists, etc., are repeatedly faced with complex and dynamic systems which

More information

Approximation Algorithms: LP Relaxation, Rounding, and Randomized Rounding Techniques. My T. Thai

Approximation Algorithms: LP Relaxation, Rounding, and Randomized Rounding Techniques. My T. Thai Approximation Algorithms: LP Relaxation, Rounding, and Randomized Rounding Techniques My T. Thai 1 Overview An overview of LP relaxation and rounding method is as follows: 1. Formulate an optimization

More information

International Doctoral School Algorithmic Decision Theory: MCDA and MOO

International Doctoral School Algorithmic Decision Theory: MCDA and MOO International Doctoral School Algorithmic Decision Theory: MCDA and MOO Lecture 2: Multiobjective Linear Programming Department of Engineering Science, The University of Auckland, New Zealand Laboratoire

More information

. P. 4.3 Basic feasible solutions and vertices of polyhedra. x 1. x 2

. P. 4.3 Basic feasible solutions and vertices of polyhedra. x 1. x 2 4. Basic feasible solutions and vertices of polyhedra Due to the fundamental theorem of Linear Programming, to solve any LP it suffices to consider the vertices (finitely many) of the polyhedron P of the

More information

Mathematics Notes for Class 12 chapter 12. Linear Programming

Mathematics Notes for Class 12 chapter 12. Linear Programming 1 P a g e Mathematics Notes for Class 12 chapter 12. Linear Programming Linear Programming It is an important optimization (maximization or minimization) technique used in decision making is business and

More information

Practical Guide to the Simplex Method of Linear Programming

Practical Guide to the Simplex Method of Linear Programming Practical Guide to the Simplex Method of Linear Programming Marcel Oliver Revised: April, 0 The basic steps of the simplex algorithm Step : Write the linear programming problem in standard form Linear

More information

Math 407A: Linear Optimization

Math 407A: Linear Optimization Math 407A: Linear Optimization Lecture 4: LP Standard Form 1 1 Author: James Burke, University of Washington LPs in Standard Form Minimization maximization Linear equations to linear inequalities Lower

More information

1 Polyhedra and Linear Programming

1 Polyhedra and Linear Programming CS 598CSC: Combinatorial Optimization Lecture date: January 21, 2009 Instructor: Chandra Chekuri Scribe: Sungjin Im 1 Polyhedra and Linear Programming In this lecture, we will cover some basic material

More information

7.1 Modelling the transportation problem

7.1 Modelling the transportation problem Chapter Transportation Problems.1 Modelling the transportation problem The transportation problem is concerned with finding the minimum cost of transporting a single commodity from a given number of sources

More information

The Graphical Simplex Method: An Example

The Graphical Simplex Method: An Example The Graphical Simplex Method: An Example Consider the following linear program: Max 4x 1 +3x Subject to: x 1 +3x 6 (1) 3x 1 +x 3 () x 5 (3) x 1 +x 4 (4) x 1, x 0. Goal: produce a pair of x 1 and x that

More information

Fundamentals of Operations Research Prof.G. Srinivasan Department of Management Studies Lecture No. # 03 Indian Institute of Technology, Madras

Fundamentals of Operations Research Prof.G. Srinivasan Department of Management Studies Lecture No. # 03 Indian Institute of Technology, Madras Fundamentals of Operations Research Prof.G. Srinivasan Department of Management Studies Lecture No. # 03 Indian Institute of Technology, Madras Linear Programming solutions - Graphical and Algebraic Methods

More information

The Simplex Method. What happens when we need more decision variables and more problem constraints?

The Simplex Method. What happens when we need more decision variables and more problem constraints? The Simplex Method The geometric method of solving linear programming problems presented before. The graphical method is useful only for problems involving two decision variables and relatively few problem

More information

Math 120 Final Exam Practice Problems, Form: A

Math 120 Final Exam Practice Problems, Form: A Math 120 Final Exam Practice Problems, Form: A Name: While every attempt was made to be complete in the types of problems given below, we make no guarantees about the completeness of the problems. Specifically,

More information

Chapter 1, Operations Research (OR)

Chapter 1, Operations Research (OR) Chapter 1, Operations Research (OR) Kent Andersen February 7, 2007 The term Operations Research refers to research on operations. In other words, the study of how to operate something in the best possible

More information

5.1 Bipartite Matching

5.1 Bipartite Matching CS787: Advanced Algorithms Lecture 5: Applications of Network Flow In the last lecture, we looked at the problem of finding the maximum flow in a graph, and how it can be efficiently solved using the Ford-Fulkerson

More information

Converting a Linear Program to Standard Form

Converting a Linear Program to Standard Form Converting a Linear Program to Standard Form Hi, welcome to a tutorial on converting an LP to Standard Form. We hope that you enjoy it and find it useful. Amit, an MIT Beaver Mita, an MIT Beaver 2 Linear

More information

Linear Programming. Solving LP Models Using MS Excel, 18

Linear Programming. Solving LP Models Using MS Excel, 18 SUPPLEMENT TO CHAPTER SIX Linear Programming SUPPLEMENT OUTLINE Introduction, 2 Linear Programming Models, 2 Model Formulation, 4 Graphical Linear Programming, 5 Outline of Graphical Procedure, 5 Plotting

More information

7.4 Linear Programming: The Simplex Method

7.4 Linear Programming: The Simplex Method 7.4 Linear Programming: The Simplex Method For linear programming problems with more than two variables, the graphical method is usually impossible, so the simplex method is used. Because the simplex method

More information

max cx s.t. Ax c where the matrix A, cost vector c and right hand side b are given and x is a vector of variables. For this example we have x

max cx s.t. Ax c where the matrix A, cost vector c and right hand side b are given and x is a vector of variables. For this example we have x Linear Programming Linear programming refers to problems stated as maximization or minimization of a linear function subject to constraints that are linear equalities and inequalities. Although the study

More information

Lecture 11: 0-1 Quadratic Program and Lower Bounds

Lecture 11: 0-1 Quadratic Program and Lower Bounds Lecture : - Quadratic Program and Lower Bounds (3 units) Outline Problem formulations Reformulation: Linearization & continuous relaxation Branch & Bound Method framework Simple bounds, LP bound and semidefinite

More information

Proximal mapping via network optimization

Proximal mapping via network optimization L. Vandenberghe EE236C (Spring 23-4) Proximal mapping via network optimization minimum cut and maximum flow problems parametric minimum cut problem application to proximal mapping Introduction this lecture:

More information

Comparison of Optimization Techniques in Large Scale Transportation Problems

Comparison of Optimization Techniques in Large Scale Transportation Problems Journal of Undergraduate Research at Minnesota State University, Mankato Volume 4 Article 10 2004 Comparison of Optimization Techniques in Large Scale Transportation Problems Tapojit Kumar Minnesota State

More information

4.6 Linear Programming duality

4.6 Linear Programming duality 4.6 Linear Programming duality To any minimization (maximization) LP we can associate a closely related maximization (minimization) LP. Different spaces and objective functions but in general same optimal

More information

Standard Form of a Linear Programming Problem

Standard Form of a Linear Programming Problem 494 CHAPTER 9 LINEAR PROGRAMMING 9. THE SIMPLEX METHOD: MAXIMIZATION For linear programming problems involving two variables, the graphical solution method introduced in Section 9. is convenient. However,

More information

Lecture notes 1: Introduction to linear and (mixed) integer programs

Lecture notes 1: Introduction to linear and (mixed) integer programs Lecture notes 1: Introduction to linear and (mixed) integer programs Vincent Conitzer 1 An example We will start with a simple example. Suppose we are in the business of selling reproductions of two different

More information

A couple of Max Min Examples. 3.6 # 2) Find the maximum possible area of a rectangle of perimeter 200m.

A couple of Max Min Examples. 3.6 # 2) Find the maximum possible area of a rectangle of perimeter 200m. A couple of Max Min Examples 3.6 # 2) Find the maximum possible area of a rectangle of perimeter 200m. Solution : As is the case with all of these problems, we need to find a function to minimize or maximize

More information

Optimization in R n Introduction

Optimization in R n Introduction Optimization in R n Introduction Rudi Pendavingh Eindhoven Technical University Optimization in R n, lecture Rudi Pendavingh (TUE) Optimization in R n Introduction ORN / 4 Some optimization problems designing

More information

3. Linear Programming and Polyhedral Combinatorics

3. Linear Programming and Polyhedral Combinatorics Massachusetts Institute of Technology Handout 6 18.433: Combinatorial Optimization February 20th, 2009 Michel X. Goemans 3. Linear Programming and Polyhedral Combinatorics Summary of what was seen in the

More information

4 UNIT FOUR: Transportation and Assignment problems

4 UNIT FOUR: Transportation and Assignment problems 4 UNIT FOUR: Transportation and Assignment problems 4.1 Objectives By the end of this unit you will be able to: formulate special linear programming problems using the transportation model. define a balanced

More information

An Introduction to Linear Programming

An Introduction to Linear Programming An Introduction to Linear Programming Steven J. Miller March 31, 2007 Mathematics Department Brown University 151 Thayer Street Providence, RI 02912 Abstract We describe Linear Programming, an important

More information

Assignment Problems. Guoming Tang

Assignment Problems. Guoming Tang Assignment Problems Guoming Tang A Practical Problem Workers: A, B, C, D Jobs: P, Q, R, S. Cost matrix: Job P Job Q Job R Job S Worker A 1 2 3 4 Worker B 2 4 6 8 Worker C 3 6 9 12 Worker D 4 8 12 16 Given:

More information

princeton univ. F 13 cos 521: Advanced Algorithm Design Lecture 6: Provable Approximation via Linear Programming Lecturer: Sanjeev Arora

princeton univ. F 13 cos 521: Advanced Algorithm Design Lecture 6: Provable Approximation via Linear Programming Lecturer: Sanjeev Arora princeton univ. F 13 cos 521: Advanced Algorithm Design Lecture 6: Provable Approximation via Linear Programming Lecturer: Sanjeev Arora Scribe: One of the running themes in this course is the notion of

More information

Lesson 3: Graphical method for solving LPP. Graphical Method of solving Linear Programming Problems

Lesson 3: Graphical method for solving LPP. Graphical Method of solving Linear Programming Problems Unit Lesson 3: Graphical method for solving LPP. Learning outcome.finding the graphical solution to the linear programming model Graphical Method of solving Linear Programming Problems Introduction Dear

More information

Actually Doing It! 6. Prove that the regular unit cube (say 1cm=unit) of sufficiently high dimension can fit inside it the whole city of New York.

Actually Doing It! 6. Prove that the regular unit cube (say 1cm=unit) of sufficiently high dimension can fit inside it the whole city of New York. 1: 1. Compute a random 4-dimensional polytope P as the convex hull of 10 random points using rand sphere(4,10). Run VISUAL to see a Schlegel diagram. How many 3-dimensional polytopes do you see? How many

More information

Optimization of Design. Lecturer:Dung-An Wang Lecture 12

Optimization of Design. Lecturer:Dung-An Wang Lecture 12 Optimization of Design Lecturer:Dung-An Wang Lecture 12 Lecture outline Reading: Ch12 of text Today s lecture 2 Constrained nonlinear programming problem Find x=(x1,..., xn), a design variable vector of

More information

Lecture 2: The SVM classifier

Lecture 2: The SVM classifier Lecture 2: The SVM classifier C19 Machine Learning Hilary 2015 A. Zisserman Review of linear classifiers Linear separability Perceptron Support Vector Machine (SVM) classifier Wide margin Cost function

More information

Lecture 7: Approximation via Randomized Rounding

Lecture 7: Approximation via Randomized Rounding Lecture 7: Approximation via Randomized Rounding Often LPs return a fractional solution where the solution x, which is supposed to be in {0, } n, is in [0, ] n instead. There is a generic way of obtaining

More information

Largest Fixed-Aspect, Axis-Aligned Rectangle

Largest Fixed-Aspect, Axis-Aligned Rectangle Largest Fixed-Aspect, Axis-Aligned Rectangle David Eberly Geometric Tools, LLC http://www.geometrictools.com/ Copyright c 1998-2016. All Rights Reserved. Created: February 21, 2004 Last Modified: February

More information

3 Does the Simplex Algorithm Work?

3 Does the Simplex Algorithm Work? Does the Simplex Algorithm Work? In this section we carefully examine the simplex algorithm introduced in the previous chapter. Our goal is to either prove that it works, or to determine those circumstances

More information

LECTURE: INTRO TO LINEAR PROGRAMMING AND THE SIMPLEX METHOD, KEVIN ROSS MARCH 31, 2005

LECTURE: INTRO TO LINEAR PROGRAMMING AND THE SIMPLEX METHOD, KEVIN ROSS MARCH 31, 2005 LECTURE: INTRO TO LINEAR PROGRAMMING AND THE SIMPLEX METHOD, KEVIN ROSS MARCH 31, 2005 DAVID L. BERNICK dbernick@soe.ucsc.edu 1. Overview Typical Linear Programming problems Standard form and converting

More information

4. How many integers between 2004 and 4002 are perfect squares?

4. How many integers between 2004 and 4002 are perfect squares? 5 is 0% of what number? What is the value of + 3 4 + 99 00? (alternating signs) 3 A frog is at the bottom of a well 0 feet deep It climbs up 3 feet every day, but slides back feet each night If it started

More information

Chapter 3 LINEAR PROGRAMMING GRAPHICAL SOLUTION 3.1 SOLUTION METHODS 3.2 TERMINOLOGY

Chapter 3 LINEAR PROGRAMMING GRAPHICAL SOLUTION 3.1 SOLUTION METHODS 3.2 TERMINOLOGY Chapter 3 LINEAR PROGRAMMING GRAPHICAL SOLUTION 3.1 SOLUTION METHODS Once the problem is formulated by setting appropriate objective function and constraints, the next step is to solve it. Solving LPP

More information

Nonlinear Programming Methods.S2 Quadratic Programming

Nonlinear Programming Methods.S2 Quadratic Programming Nonlinear Programming Methods.S2 Quadratic Programming Operations Research Models and Methods Paul A. Jensen and Jonathan F. Bard A linearly constrained optimization problem with a quadratic objective

More information

Notes on Matrix Multiplication and the Transitive Closure

Notes on Matrix Multiplication and the Transitive Closure ICS 6D Due: Wednesday, February 25, 2015 Instructor: Sandy Irani Notes on Matrix Multiplication and the Transitive Closure An n m matrix over a set S is an array of elements from S with n rows and m columns.

More information

Lecture 2. Marginal Functions, Average Functions, Elasticity, the Marginal Principle, and Constrained Optimization

Lecture 2. Marginal Functions, Average Functions, Elasticity, the Marginal Principle, and Constrained Optimization Lecture 2. Marginal Functions, Average Functions, Elasticity, the Marginal Principle, and Constrained Optimization 2.1. Introduction Suppose that an economic relationship can be described by a real-valued

More information

6. Mixed Integer Linear Programming

6. Mixed Integer Linear Programming 6. Mixed Integer Linear Programming Javier Larrosa Albert Oliveras Enric Rodríguez-Carbonell Problem Solving and Constraint Programming (RPAR) Session 6 p.1/40 Mixed Integer Linear Programming A mixed

More information

4.4 The Simplex Method: Non-Standard Form

4.4 The Simplex Method: Non-Standard Form 4.4 The Simplex Method: Non-Standard Form Standard form and what can be relaxed What were the conditions for standard form we have been adhering to? Standard form and what can be relaxed What were the

More information

Theory of Linear Programming

Theory of Linear Programming Theory of Linear Programming Debasis Mishra April 6, 2011 1 Introduction Optimization of a function f over a set S involves finding the maximum (minimum) value of f (objective function) in the set S (feasible

More information

Linear Programming with Post-Optimality Analyses

Linear Programming with Post-Optimality Analyses Linear Programming with Post-Optimality Analyses Wilson Problem: Wilson Manufacturing produces both baseballs and softballs, which it wholesales to vendors around the country. Its facilities permit the

More information

Linear Inequalities and Linear Programming. Systems of Linear Inequalities in Two Variables

Linear Inequalities and Linear Programming. Systems of Linear Inequalities in Two Variables Linear Inequalities and Linear Programming 5.1 Systems of Linear Inequalities 5.2 Linear Programming Geometric Approach 5.3 Geometric Introduction to Simplex Method 5.4 Maximization with constraints 5.5

More information

26 Linear Programming

26 Linear Programming The greatest flood has the soonest ebb; the sorest tempest the most sudden calm; the hottest love the coldest end; and from the deepest desire oftentimes ensues the deadliest hate. Th extremes of glory

More information

Components of LP Problem. Chapter 13. Linear Programming (LP): Model Formulation & Graphical Solution. Introduction. Components of LP Problem (Cont.

Components of LP Problem. Chapter 13. Linear Programming (LP): Model Formulation & Graphical Solution. Introduction. Components of LP Problem (Cont. Chapter 13 Linear Programming (LP): Model Formulation & Graphical Solution Components of LP Problem Decision Variables Denoted by mathematical symbols that does not have a specific value Examples How much

More information

Decision Mathematics D1. Advanced/Advanced Subsidiary. Friday 12 January 2007 Morning Time: 1 hour 30 minutes. D1 answer book

Decision Mathematics D1. Advanced/Advanced Subsidiary. Friday 12 January 2007 Morning Time: 1 hour 30 minutes. D1 answer book Paper Reference(s) 6689/01 Edexcel GCE Decision Mathematics D1 Advanced/Advanced Subsidiary Friday 12 January 2007 Morning Time: 1 hour 30 minutes Materials required for examination Nil Items included

More information

Mathematical finance and linear programming (optimization)

Mathematical finance and linear programming (optimization) Mathematical finance and linear programming (optimization) Geir Dahl September 15, 2009 1 Introduction The purpose of this short note is to explain how linear programming (LP) (=linear optimization) may

More information

A counter-example to the Hirsch conjecture

A counter-example to the Hirsch conjecture A counter-example to the Hirsch conjecture Francisco Santos Universidad de Cantabria, Spain http://personales.unican.es/santosf/hirsch Bernoulli Conference on Discrete and Computational Geometry Lausanne,

More information